Lesson 4: Exponential Growth and Decay

Property

For exponential growth and decay functions of the form $f(x) = a \cdot b^x$:

• When $b > 1$, the function represents exponential growth and the graph is increasing.
• When $0 < b < 1$, the function represents exponential decay and the graph is decreasing.
• The initial value $a$ determines the y-intercept at $(0, a)$.
• The x-axis ($y = 0$) is a horizontal asymptote for all exponential functions.

Examples

Property

To write an exponential function from given information:
For growth: $y = a(1 + r)^t$ where $a$ is initial value and $r$ is growth rate as decimal
For decay: $y = a(1 - r)^t$ where $a$ is initial value and $r$ is decay rate as decimal

Examples

Property

To convert a growth rate percentage to a growth factor: Growth Factor = $1 + \frac{\text{growth rate percent}}{100}$
For exponential growth function $y = a(1 + r)^t$, the growth factor is $(1 + r)$ where $r$ is the decimal form of the growth rate.

Examples

Property

Exponential functions can be rewritten using exponent properties to reveal different time periods or rates. The key property is $(a^m)^n = a^{mn}$, which allows us to transform $y = a(b)^t$ into equivalent forms like $y = a(b^{12})^{\frac{t}{12}}$ to show monthly rates from annual rates.

Examples